Spin and orbital Yu-Shiba-Rusinov states and delocalization control of magnetic order in parallel double quantum dot Josephson junctions

With the help of the numerical renormalization group method, we theoretically investigate the Josephson phase transition and the Yu-Shiba-Rusinov (YSR) states in parallel-coupled double quantum dot (QD) Josephson junctions. The interplay of competition and cooperation of multiple energy scales within the system results in distinct impacts on the 0-$\ensuremath{\pi}$ transition behavior and the evolution of YSR states when the QDs are in the half-filled case. This arises from the relative variation in the strengths of the interdot antiferromagnetic correlation and the Ruderman-Kittel-Kasuya-Yosida interaction. In addition, when the interdot coupling coefficient ${t}_{12}$ is fixed, by adjusting the phase difference between superconducting electrodes, we found that the magnetic order of two QDs and the YSR subgap states can be controlled in a delocalized manner. Our research explores the complexities of electron correlation effects in Josephson junctions, providing unique insights into the construction and regulation of superconducting nanodevices and quantum bits in the future. Furthermore, the system's spin and orbital degrees of freedom can be manipulated to satisfy the SU(4) symmetry by adjusting the QD levels and other parameters. An orbital YSR state is induced by a pseudospin-flip process involving pseudospins and quasiparticles outside the superconducting gap. We provide a detailed analysis of the various physical mechanisms underlying the emergence of spin and orbital YSR states through the application of an external magnetic field.